Another look at the trinomial of the form: $ax^n+bx+c=0$

Question

Has the trinomial of the form $ax^n+bx+c=0$ been fully studied for $n>2$? If so, please let me know of any reference or interesting findings. Thanks.

Answer 1

Yes, the equation,

$$ax^p+bx^q+c = 0\tag1$$

has been studied in detail and can be solved in terms of hypergeometric functions in one variable. To quote (eq.42) of this Mathworld entry,

"... This technique gives closed-form solutions in terms of hypergeometric functions in one variable for any polynomial equation which can be written in the form $x^p+bx^q+c = 0$..."

P.S. If in radicals, there are special cases that can be so solved, such as the irreducible,

$$x^5-5x^2-3 = 0$$

$$x^6+3x+3 = 0$$

$$x^8-5x-5=0$$

There are an infinite number of solvable examples for $p=5$ and, if I remember correctly, for $p=6,8$ as well. However, none are known for $p=7$.